homework #1 solutions

The objectives of this homework assignment are the followings:

  • Build your own RMarkdown document.
  • Master different aspects of RMarkdown syntax.
  • Become familiar with GitHub as a collaborative tool.

Note the following requirements:

  • All members of the group must commit at least once.
  • All commit messages must be reasonably clear and meaningful.
  • Your GitHub repository must include at least one issue containing some form of TO DO list.

In your repository, create a RMarkdown file called hw1.Rmd providing an HTML output with the theme cerulean and syntax highlighting tango. This file should contain the following elements:

  • A “title” section which should at least include:
    • A title (e.g. Homework 1)
    • The authors
    • The date (think of using Sys.time())

In order to do so, it is possible to simply write as a header to your RMarkdown:

---
title: "Homework 1"
author: Samuel Orso \& Iegor Rudnytskyi
date: "`r format(Sys.time(), '%d %B, %Y')`"
output: html_document:
         theme: cerulean
         highlight: tango
---
  • A section called “Introduction” where you provide a short summary of the structure of your homework. Moreover, record a short video to introduce your group and include it in your RMarkdown document.

To create a section, use a single hash sign ("#"); to create a subsection, use a double hash sign ("##"). Here, one would write, for the introduction:

# Introduction 
Here is the intro. Bla bla bla.

To add a video below, one would add the following nomenclature, for example for a Youtube video:

![](https://youtu.be/zNzZ1PfUDNk)
  • A section called “Group Members”. This section should have one sub-section for each group member in your team. For example, a group with three members should have three sub-sections. Each of these sub-sections (named after each group member) should include small biographies containing at least the following elements:
    • A picture of your choice (preferably of yourself). Make sure to include a caption for this image.
    • A paragraph describing your favorite hobby as well as one interesting fact about yourself (preferably true).
    • Your favorite quote in blockquote format. Make sure to reference your quote using BibTex.
    • A table having two columns (first column containing the classes you are following this semester; second column containing the time of these classes).

To do the above, one would for example have the following nomenclature, using LaTeX type format for the table:

# Group Members
## Samuel Orso
Lecturer in Data Science at UNIL,...
![An American logo](http://www.alumni.psu.edu/s/1218/16/images/logo.png)

My hobbies are... 

>"Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it."
>-- Tolstoi

To create a table, one could for example use the kableExtra package and write:

library(knitr)
library(kableExtra)
courses.taken <- c("Applied Mathematics", "Measure Theory", "R for Data Science")
time <- c("08:00", "08:00", "09:00")
df <- data.frame(cbind(courses.taken, time))
colnames(df) <- c("Courses taken", "Time")
kable_styling(kable(df, "html"))
  • A section called “RMarkdown Syntax”, where you will demonstrate your RMarkdown skills! In this section make sure to:
    • Show an example where the chunk option cache = T leads to a misleading answer. This example must be different from the one presented in the textbook.

For example, this can be use to save some time while generating large random vectors. One example where cache=TRUE can be misleading is when you first start by a chunk of the following form, specifying for the option:

set.seed(1)
a <- rexp(100)

and you call a in another chunk, say the following one, with option cache=TRUE:

(b <- log(a))

without adding as option dependson=(name of the first chunk), since if you first knit with 100 randomly generated exponential numbers and you then change a to rexp(1000), it will cache the value of a and only show you b accordingly.

  • Simulate 300 random samples from a distribution that is uniform on [-1, 1] interval using the function runif(). Store these 300 values in a vector called x. Then, compute the empirical median, mean and variance of x. Are these results different from 0, 0 and 1/3 (their respective theoretical values)? Is this result surprising? Justify your answer.

Here is the code to do so:

set.seed(1)
x <- runif(n = 300, min = -1, max = 1)
empmean <- mean(x)
empmed <- median(x)
empvar <- var(x)

The values are somewhat different from 0, 0 and 1/3 respectively, not so much but this variation is due to the fact that we only compute sample estimates of population values for the values above. Therefore, there is a natural variation.

  • Include a graph showing the density estimation of x (make sure to include a caption to this figure). You can use the R function d <- denstiy() with default parameters, and the plot(d) function. Why does the estimated density exceed the [-1, 1] interval?

One would use the following command:

d <- density(x)
plot(d, main = "Density")

adding as an option to this chunk the option fig.cap="Density estimation of x" to display a caption to this figure.

By default density estimator uses “gaussian” kernel to construct the curve. Therefore, the points that are close to the borders (-1 and 1) add extra noise, which expands outside of the original interval. This noise may be reduced by decreasing the bandwidth bw, as well as choosing a kernel with finite support, i.e. rectangular or epanechnikov.

  • Include the following equation eq:
  
  \[
  \begin{aligned}
      \left( P^t f \right) ( \mathbf{x} ) = 
      f_b ( \mathbf{x} ) +
      \frac{\sigma^2 t}{2 |D| d} \int_{\partial D}
      \frac{\partial f}{\partial n} \mathrm{d} \widehat{\mathbf{y}} +
      \sum c_m^0 \exp \left(
      - \frac{\sigma^2 \widetilde{\kappa}_m^2 t}{2}
      \right)
      \widetilde{s}_m ( \mathbf{x} )
      + f_h ( \mathbf{x} )
  \end{aligned}
  \]
  • Include the following in-line equation:

To do so, use the LaTeX nomenclature as follow:

$\hat{f}(\xi) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) \, e^{-i x \xi} \, \mathrm{d} x.$
  • Include the following text in blue: “The degree of civilization in a society can be judged by entering its prisons.”, Fyodor Dostoevsky

Write:

<span style="color:green">"The degree of civilization in a society can be judged by entering its prisons.", Fyodor Dostoevsky text</span>
  • Include a “More info” button with hide/unhide functionality.

To do so, use:

  <button data-toggle="collapse" data-target="#demo">More info</button>
  <div id="demo" class="collapse">
    Some additional info... 
  </div>
  • Include a “color box”" with some text.

To do so, use:

  <div class="alert alert-danger">
    <strong>Some important Info:</strong> something
  </div>